A unified approach to the STFT, TFDs, and instantaneous frequency

Cohen's class of time-frequency representations (TFRs) is reformulated into a discrete-time, discrete-frequency, computer-implementable form. It is shown how, in this form, many of the properties of the continuous-time, continuous-frequency formulation are either lost or altered. Intuitions applicable in the continuous-time case do not necessarily carry over to the discrete-time case examined. The properties of the discrete variable formulation examined are the presence and form of cross-terms, instantaneous frequency estimation, and relationships between Cohen's class of TFRs. A parameterized class of distributions which is a blending between the short-time Fourier transform (STFT) and the Wigner-Ville distribution. The two main conclusions are that all TFRs of Cohen's class implementable in the given form (which includes all commonly used TFRs) possess cross-terms and that instantaneous frequency estimation using periodic moments of these TFRs is purposeless, since simpler methods obtain the same result     

On the equivalence of the operator and kernel methods for jointdistributions of arbitrary variables

Generalizing the concept of time-frequency representations, Cohen (see Englewood Cliffs, NJ: Prentice-Hall, 1995) has proposed a method, based on operator correspondence rules, for generating joint distributions of arbitrary variables. As an alternative to considering all such rules, which is a practical impossibility in general, Cohen has proposed the kernel method in which different distributions are generated from a fixed rule via an arbitrary kernel. We derive a simple but rather stringent necessary condition, on the underlying operators, for the kernel method (with the kernel functionally independent of the variables) to generate all bilinear distributions. Of the specific pairs of variables that have been studied, essentially only time and frequency satisfy the condition; in particular, the important variables of time and scale do not. The results warrant further study for a systematic characterization of bilinear distributions in Cohen's method     

On alias-free formulations of discrete-time Cohen's class ofdistributions

The transition of the Cohen's (1989) class of distributions from the continuous-time case to the discrete-time case is not straightforward because of aliasing problems. We classify the aliasing problems, which occur for joint time-frequency representations (TFRs), into two categories: type-I and type-II aliasings. Type-I aliasing can be avoided by properly defined discrete-time versions of some members of Cohen's class (in particular, properly defined kernels), whereas type-II aliasing can be reduced and/or eliminated by increasing the sampling rate. A type-I alias-free formulation of the discrete-time Cohen's class (AF-DTCC), which is equivalent to the AF-GDTFT of Joeng and Williams (see ibid., vol.40, no.2, p.1084, 1992) is then introduced based on the fact that the Cohen's class can be expressed as the 2-D Fourier transform of the generalized ambiguity function (AF). Based on this definition, two discretization schemes for kernel functions are presented in both the AF domain and the time-lag domain, and are shown to be equivalent under certain conditions. We also do the following: (1) we show that a discrete-time Wigner-Ville distribution (DWVD) and discrete-time spectrogram (DSPG) are type-I alias-free and members of AF-DTCC; (2) we use all the available correlation information from a given data sequence by using the Woodward AF instead of the Sussman AF; (3) we give kernel constraints in the AF domain for various distribution properties; and (4) we provide a type-I and type-II alias-free formulation for those distributions whose kernel functions satisfy the finite frequency-support constraint     

Beyond time-frequency analysis: energy densities in one and manydimensions

Given a unitary operator A representing a physical quantity of interest, we employ concepts from group representation theory to define two natural signal energy densities for A. The first is invariant to A and proves useful when the effect of A is to be ignored; the second is covariant to A and measures the “A” content of signals. We also consider joint densities for multiple operators and, in the process, provide an alternative interpretation of Cohen's (see Englewood Cliffs, NJ: Prentice-Hall, 1995) general construction for joint distributions of arbitrary variables     

Kernel design for reduced interference distributions

The authors present a class of time-frequency signal representations (TFRs) called the reduced interference distribution (RID). An overview of commonly used TFRs is given, and desirable distribution properties are introduced. Particular attention is paid to the interpretation of Cohen's class of time-frequency distributions of TFRs in the ambiguity, temporal correlation, spectral correlation, and time-frequency domains. Based on the desirable kernel requirements, the RID is discussed and further defined. A systematic procedure to create RID kernels, (or, equivalently, compute RIDs) is proposed. Some aspects and properties of the RID are discussed. The authors estimate design considerations for RIDs and compare various selections of the primitive window. Some experimental results demonstrating the performance of the RID are presented     

Vector-valued time-frequency representations

Cohen's (1989) class of time frequency distributions (TFDs), which includes the spectrogram (SP), Wigner distribution (WD), and reduced interference distributions (RIDs) has become widely known as a useful signal analysis tool. It has been shown that every real-valued TFD can be written as a weighted sum of SPs. The “SP decomposition” has been used to construct fast approximations to desirable TFDs using the SP building block, for which there exist accessible and efficient hardware and software implementations. We introduce a class of linear, vector-valued time-frequency representations (TFRs) that are easily related to associated bilinear TFDs through the SP decomposition. We solve a least-squares signal synthesis problem on modified vector-valued TFRs that are associated with nonnegative TFDs as a weighted sum of least-squares short-time Fourier transform (STFT) signal synthesis schemes. We extend the solution to vector-valued TFRs associated with high-resolution TFDs in order to define a high-resolution alternative to STFT signal synthesis, as demonstrated by desirable properties and examples. The resulting signal synthesis methods can be realized as a weighted sum of STFT synthesis schemes, for which there exist accessible and efficient hardware and software implementations     

The statistical performance of some instantaneous frequencyestimators

The authors examine the class of smoothed central finite difference (SCFD) instantaneous frequency (IF) estimators which are based on finite differencing of the phase of the analytic signal. These estimators are closely related to IF estimation via the (periodic) first moment, with respect to frequency of discrete time-frequency representations (TFRs) in L. Cohen's (1966) class. The authors determine the distribution of this class of estimators and establish a framework which allows the comparison of several other estimators such as the zero-crossing estimator and one based on linear regression on the signal phase. It is found that the regression IF estimator is biased and exhibits a large threshold for much of the frequency range. By replacing the linear convolution operation in the regression estimator with the appropriate convolution operation for circular data the authors obtain the parabolic SCFD (PSCFD) estimator, which is unbiased and has a frequency-independent variance, yet retains the optimal performance and simplicity of the original estimator     

A general approach for obtaining joint representations in signalanalysis. I. Characteristic function operator method

A general approach for obtaining joint representations for arbitrary physical quantities is presented. The characteristic function operator method of Moyal (1949) and Ville (1960) and generalized by Cohen (1966, 1976) and Scully and Cohen (1987) for the time-frequency case is developed for arbitrary variables     

二次时频表示中核函数的优化设计

二次时频分布是分析非平稳信号的有力工具，在具有许多优良特性的同时，存在严重的交叉干扰项。在Wigner-Ville分布及Cohen类时频分布具有固定核函数的基础上，研究了基于信号的核函数优化设计的两种方法，径向高斯核函数和最优相位核函数的设计方法。基于信号的核函数的时频表示可以有效地抑制或转移交叉分量，提高时频表示的可读估计，改善其主要性能。     

Integral transforms covariant to unitary operators and theirimplications for joint signal representations

Fundamental to the theory of joint signal representations is the idea of associating a variable, such as time or frequency, with an operator, a concept borrowed from quantum mechanics. Each variable can be associated with a Hermitian operator, or equivalently and consistently, as we show, with a parameterized unitary operator. It is well known that the eigenfunctions of the unitary operator define a signal representation which is invariant to the effect of the unitary operator on the signal, and is hence useful when such changes in the signal are to be ignored. However, for detection or estimation of such changes, a signal representation covariant to them is needed. Using well-known results in functional analysis, we show that there always exists a translationally covariant representation; that is, an application of the operator produces a corresponding translation in the representation. This is a generalization of a recent result in which a transform covariant to dilations is presented. Using Stone's theorem, the “covariant” transform naturally leads to the definition of another, unique, dual parameterized unitary operator. This notion of duality, which we make precise, has important implications for joint distributions of arbitrary variables and their interpretation. In particular, joint distributions of dual variables are structurally equivalent to Cohen's class of time-frequency representations, and our development shows that, for two variables, the Hermitian and unitary operator correspondences can be used consistently and interchangeably if and only if the variables are dual     

Alias-free generalized discrete-time time-frequency distributions

A definition of generalized discrete-time time-frequency distribution that utilizes all of the outer product terms from a data sequence, so that one can avoid aliasing, is introduced. The new approach provides (1) proper implementation of the discrete-time spectrogram, (2) correct evaluation of the instantaneous frequency of the underlying continuous-time signal, and (3) correct frequency marginal. The formulation provides a unified framework for implementing members of Cohen's class, which was formulated in the continuous-time domain. Some requirements for the discrete-time kernel in the new approach are discussed in association with desirable distribution properties. Some experimental results are provided to illustrate the features of the proposed method     

Optimal quadratic detection and estimation using generalized jointsignal representations

Time-frequency analysis has significant advances in two main directions: statistically optimized methods that extend the scope of time-frequency-based techniques from merely exploratory data analysis to more quantitative application and generalized joint signal representations that extend time-frequency-based methods to a richer class of nonstationary signals. This paper fuses the two advances by developing optimal detection and estimation techniques based on generalized joint signal representations. By generalizing the statistical methods developed for time-frequency representations to arbitrary joint signal representations, this paper develops a unified theory applicable to a wide variety of problems in nonstationary statistical signal processing     

Measuring time-frequency information content using the Renyientropies

The generalized entropies of Renyi inspire new measures for estimating signal information and complexity in the time-frequency plane. When applied to a time-frequency representation (TFR) from Cohen's class or the affine class, the Renyi entropies conform closely to the notion of complexity that we use when visually inspecting time-frequency images. These measures possess several additional interesting and useful properties, such as accounting and cross-component and transformation invariances, that make them natural for time-frequency analysis. This paper comprises a detailed study of the properties and several potential applications of the Renyi entropies, with emphasis on the mathematical foundations for quadratic TFRs. In particular, for the Wigner distribution, we establish that there exist signals for which the measures are not well defined     

基于Wigner分布时频遮隔的信号分解算法

综合特征值分解及Wigner分布时频遮隔提出了一种信号分解算法,并推广应用于其他交叉项抑制时频表示.对于由时频面上互不重叠分量合成的多分量信号,证明了信号分量可与各分量Wigner分布之和的逆Fourier变换的特征值分解相对应;通过阈值法可从抑制交叉项时频表示获得信号时频支撑区域,以此为模板遮隔Wigner分布可减少交叉项并保持自项聚集性,其逆Fourier变换的特征值分解就可实现多分量信号分解.仿真实例分析结果表明了该理论与算法的正确性和实用性.最后分析了算法性能并拓展了其实用范围.     

Design of time-frequency representations using a multiform,tiltable exponential kernel

A novel Cohen's (1981) class time-frequency representation with a tiltable, generalized exponential kernel capable of attaining a wide diversity of shapes in the ambiguity function plane is proposed for improving the time-frequency analysis of multicomponent signals. The first advantage of the proposed kernel is its ability to generate a wider variety of passband shapes, e.g., rotated ellipses, generalized hyperbolas, diamonds, rectangles, parallel strips at arbitrary angles, crosses, snowflakes, etc., and narrower transition regions than conventional Cohen's class kernels; this versatility enables the new kernel to suppress undesirable cross terms in a broader variety of time-frequency scenarios. The second advantage of the new kernel is that closed form design equations can now be easily derived to select kernel parameters that meet or exceed a given set of user specified passband and stopband design criteria in the ambiguity function plane. Thirdly, it is shown that simple constraints on the parameters of the new kernel can be used to guarantee many desirable properties of time-frequency representations. The well known Choi-Williams (1989) exponential kernel, the generalized exponential kernel, and Nuttall's (1990) tilted Gaussian kernel are special cases of the proposed kernel     

Unified study on characteristic spectrum representation of signals and spectral decomposition for stationary random signals

The unified relationship between the signal characteristic spectrum representation and the spectral decomposition for the stationary random signal was deeply studied. By using the relations among the differential operators, the integral operator and the Green's function of the characteristic differential equation, the inverse relationship between the Hermitian differential operator and the Hermitian integral operator were given, the characteristic differential equation and corresponding characteristic integral equation were demonstrated, and the spectral representations of both Hermitian differential and integral operators and the general spectral representations for both operators were provided. Based on the superposition method of the stochastic simple harmonic vibration and the Hilbert space unitary operator method for the stationary random signal spectral decomposition, the connection and unification on mathematics of the signal characteristic spectral representation and the stationary random signal spectral decomposition are revealed.     

A general approach for obtaining joint representations in signalanalysis. II. General class, mean and local values, and bandwidth

For pt.I see ibid., vol.44, no.5, p.1080-90 (1996). Using the method developed in Cohen (1996) (Part I), the concepts of instantaneous frequency and group delay are generalized to arbitrary variables; in addition, new expressions for mean values and bandwidths are obtained. The kernel method is used to define a general class for arbitrary variables. As in the time-frequency case, the general class generates all possible distributions. The method is also formulated in terms of the local autocorrelation function     

Optimizing time-frequency kernels for classification

In many pattern recognition applications, features are traditionally extracted from standard time-frequency representations (TFRs). This assumes that the implicit smoothing of, say, a spectrogram is appropriate for the classification task. Making such assumptions may degrade classification performance. In general, ana time-frequency classification technique that uses a singular quadratic TFR (e.g., the spectrogram) as a source of features will never surpass the performance of the same technique using a regular quadratic TFR (e,g., Rihaczek or Wigner-Ville). Any TFR that is not regular is said to be singular. Use of a singular quadratic TFR implicitly discards information without explicitly determining if it is germane to the classification task. We propose smoothing regular quadratic TFRs to retain only that information that is essential for classification. We call the resulting quadratic TFRs class-dependent TFRs. This approach makes no a priori assumptions about the amount and type of time-frequency smoothing required for classification. The performance of our approach is demonstrated on simulated and real data. The simulated study indicates that the performance can approach the Bayes optimal classifier. The real-world pilot studies involved helicopter fault diagnosis and radar transmitter identification     

Infinite series representations of the trivariate and quadrivariate nakagami-m distributions

In this paper, using Miller's approach and Dougall's identity, we derive new infinite series representations for the quadrivariate Nakagami-m joint density function, cumulative distribution function (cdf) and characteristic functions (chf). The classical joint density function of exponentially correlated Nakagami-m variables can be identified as a special case of the joint density function obtained here. Our results are based on the most general arbitrary correlation matrix possible. Moreover, the trivariate density function, cdf and chf for an arbitrary correlation matrix are also derived from our main result. Bounds on the series truncation error are also presented. Finally, we develop several representative applications: the outage probability of triple branch selection combining (SC), the moments of the equal gain combining (EGC) output signal to noise ratio (SNR) and the moment generation function of the generalized SC(2,3) output SNR in an arbitrarily correlated Nakagami-m environment. Simulation results are also presented to verify the accuracy of our theoretical results.